We have revealed the geometrical origin of CP embedded into the 10D proper Lorentz
transformation with an emphasis on the complex structure of compact
6D spaces, in particular the CY threefolds.
We find that the anti-holomorphic involution of the complex structure is regarded
as the anti-holomorphic involution of period integrals on CY threefolds
with the large complex structure.
Consequently, the anti-holomorphic involution of the period integrals corresponds to
the outer automorphism of \(Sp(2h^{2,1}+2,\mathbb {Z})\)  symplectic modular group,
rather than the element of \(Sp(2h^{2,1}+2,\mathbb {Z})\)  in the complex structure moduli space.
The moduli group is then enlarged into \(Sp(2h^{2,1}+2,\mathbb {Z})\rtimes {\cal CP}\) .
That is a natural extension of the known toroidal cases, where the CP symmetry is regarded
as the outer automorphism of the \(SL(2,\mathbb {Z})\)  modular group [1]}, [2]}.
